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Structural Analysis and Design of Dynamic-Flow Networks: Implications in the Brain Dynamics

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2016 American Control Conference (ACC) Boston Marriott Copley Place July 6-8, 2016. Boston, MA, USA Structural Analysis and Design of Dynamic-Flow Networks: Implications in the Brain Dynamics Sergio´ Pequito y Ankit N. Khambhati \;[ George J. Pappas y Dragoslav D. Siljakˇ z Danielle Bassett y;\;[ Brian Litt \;[;] Abstract— In this paper, we study dynamic-flow networks, the edges are assumed constant over time. However, this is a i.e., networks described by a graph whose weights evolve rather rigid description that often fails to adequately represent according to linear differential equations. Further, these linear the structure of natural, social and man-made processes, differential equations depend on the incidence relation of the edges in a node, and possibly nodal dynamics. Because some especially in uncertain environments where the relations of these weights and their dependencies may not be accurately between the constituent parts change over time [2]. known, we extend the notion of structural controllability for Dynamic-flow networks model dynamical systems cap- dynamic-flow networks, and provide necessary and sufficient tured by networks where the weights on the edges evolve conditions for this to hold. Next, we show that the analysis according to linear differential equations. In addition, these of structural controllability in dynamic-flow networks can be reduced to that of a digraph which we refer to as meta digraph. linear differential equations depend on the incidence relation In addition, we consider different actuation capabilities, i.e., we of the edges in a node, and possibly nodal dynamics. For ex- assume that both the nodes and edges in the dynamic-flow ample, in social networks, a node (representing an individual) network can be actuated, and we explore the implications constantly processes information received from its upstream in terms of computational complexity when the minimum neighbors and makes decisions that are communicated to its cost-placement of actuators is considered. The proposed framework can be used to identify actua- downstream neighbors. The information received and passed tion capabilities required to mitigate epileptic-brain dynamics. by a node can be represented by the state variables on its More precisely, the functional connectivity of mesoscale brain incoming and outgoing edges. Thus, mapping the signals of dynamics can be modeled as a dynamic-flow network by the incoming edges onto those of the outgoing edges. This considering dynamic functional connectivity of the network. is also the case in a network of computers and routers on In the context of epilepsy, the modeling is motivated by new findings that show that the edges within seizure-generating the Internet, where the edges represent physical connections areas are almost constant over time, whereas the edges outside and the state variables on the edges represent the amount these areas exhibit higher variability over time in human epilep- of packet flow along a particular connection in a given tic networks. In addition, implementable devices to control direction. The mechanism of the nodes then corresponds to drug-resistant seizures by affecting the epileptic network has a load-balancing or routing mechanism that allows packets gained considerable attention as a viable treatment option. Subsequently, from a control-theoretic perspective, one can to reach their destination while avoiding congestion. consider actuation to attenuate edge variability responsible Dynamic-flow networks can also be used to model func- for seizure-generation in the epileptic network. In particular, tional brain networks, where nodes represent neural popula- we address the following two scenarios: (i) current placement tions and edges are statistical relationships between neural of electrical stimulators, and their probable capabilities; and activation patterns. In particular, these networks are suitable (ii) determine the minimum cost placement with minimum actuation capabilities. The latter problem is motivated by the to capture the brain dynamics when dynamic functional fact that some edges may correspond to more accessible (or less connectivity is considered [3]. More specifically, a sliding- harmful) regions in the brain, whereas others might correspond window over the blood-oxygen level dependent (BOLD) to sensitive regions in the brain. signal is considered to obtain over time the functional connectivity that captures the Pearson’s correlation between I. INTRODUCTION signals. In particular, the local dependency on adjacent edges Complex networks have traditionally been studied by is justified by the spatial-temporal dependency of the BOLD resorting to graphs [1]. These have been considered most signals [3]. of the times as static objects, whose weights associated with The control of dynamic-flow networks was introduced in [4], and in this paper we aim to account for the case This work was supported in part by the TerraSwarm Research Center, one of six centers supported by the STARnet phase of the Focus Center where the parameters describing the possible interaction Research Program (FCRP) a Semiconductor Research Corporation program dependencies in the flow-dynamic networks are either free or sponsored by MARCO and DARPA, and the NSF ECCS-1306128 grant. constant parameters over time. Towards this goal we resort zDepartment of Electrical Engineering, Santa Clara University, Santa Clara, CA 95 053, USA to structural systems theory [5] that enables the introduction yDepartment of Electrical and Systems Engineering, School of Engineer- of generic controllability notions [6], i.e., controllability ing and Applied Science, University of Pennsylvania holds for almost all possible parameterization of the free \ Department of Bioengineering, University of Pennsylvania parameters. Therefore, we aim to assess when flow-dynamic [ Penn Center for Neuroengineering and Therapeutics, University of Pennsylvania networks (seen as a dynamical system) are generically con- ] Department of Neurology, Hospital of the University of Pennsylvania trollable. In addition, we assume that different functions and 978-1-4673-8681-4/$31.00 ©2016 AACC 5758 locations of control inputs are possible, which influences the can no longer be done resorting to dual graphs, see Remark 1 edges’ weights and nodes’ dynamics. More precisely, these for additional details. ◦ can be multi-node input, i.e., the input signal is injected in The main contributions of this paper are as follows: (i) we several nodes to regulate how the dynamics of the outgoing formally introduce the notion of generic controllability for edges changes, and multi-edge inputs, i.e., an input can dynamic-flow networks, where some edges’ weights can be actuate a linear combination of edges, not necessarily the constant over time; (ii) we introduce necessary and sufficient ones that share vertices. Particular cases of these two are the conditions that ensure structural controllability; (iii) we ad- out-node inputs, i.e., all the outgoing edges from a single dress the problem of minimum actuator placement incurring vertex are driven by a scalar input signal, and the in-edge in the minimum cost while ensuring structural controllability; inputs, i.e., an input can actuate only a single edge dynamics (iv) we show how dynamic-flow networks can identify brain directly. In this paper, we analyze and provide necessary regions that should be actuated to ensure that edges’ weights and sufficient conditions for dynamic-flow networks, where variations are kept within certain bounds; (v) we show that parameters can be either free or constant over time, to current actuation capabilities are enough to ensure structural be generically controllable. In addition, we consider the controllability; and (vi) using real data, we determine the control placement problems under different assumptions; location of actuators to regulate the dynamics such that more precisely, we consider that different edges and/or nodes epileptic seizures may be attenuated and/or overcome. may incur in different costs. The rest of the paper is organized as follows. In Sec- tion II, we provide the formal problem statement. Section III Related Work reviews some concepts from computational complexity and Several necessary and sufficient conditions that character- graph theoretical concepts used in structural systems theory. ize the structural controllability, as well as their verification, Section IV presents the main technical results, followed are known for linear time-invariant [5], or switching sys- by a case study in Section V showing the implications tems [7]. Nevertheless, the selection, commonly referred to in the brain dynamics; more specifically, in the context of as design, of minimum actuation capabilities to ensure that epilepsy. Conclusions and discussions on further research are these conditions hold has only been addressed in the last presented in Section VI. years. In the context of linear time-invariant systems, the problem of determining the minimum number of actuated II. PROBLEM STATEMENT variables ensuring structural controllability was addressed in [8], [9]. Later, it was extended to the case where the mini- Let D = (V; E) be a digraph, where V denotes a set of n mum cost is sought when the actuated state variables incur in vertices, and E the set of m directed edges been the vertices. In addition, consider that each edge in the digraph has weight a cost that does not depend on the actuator considered [10], + and to the minimum cost problem allowing the actuation cost w associated with it, where w : E! R0 , and a label µ(e) 2 of a state variable to depend on the actuator considerez [11]. f1;?g, for e 2 E, where µ(e) = 1 indicates that the weight An alternative formulation consists in assuming a predefined is constant over time and µ(e) = ? indicates that it varies collection of actuators from which one selects those to be over time. Further, let N = fNigi2I represent the multi- used to ensure structural controllability, but in such scenario node actuation signal, where the collection of the nodes’ set the problem of determining the minimum collection of such Ni ⊂ f1; : : : ; ng is actuated by the actuator i. In particular, actuators is NP-hard [12].
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